Fundamentals of Plasma Physics

330 Chapter 10. Stability of static MHD equilibria

here the circumflex means the modified Bessel function is evaluated at the wall, i.e.,

with its argument set to|k|b.Equation (10.159) can then be recast as

χ=β

[

I|m|(|k|r)Kˆ|′m|−Iˆ|′m|K|m|(|k|r)

]

(10.162)

whereβ=β 1 /Kˆ|′m|.The wall boundary condition has reduced the number of inde-

pendent coefficients by one.

2. Vacuum side of plasma-vacuum interface: Here Eq.(10.147) is linearized to obtain

B 1 ·∇S 0 +B 0 ·∇S 1 =0 (10.163)

or using Eqs.(10.145) and (10.146)

B ̄ 1 vr−i(m

a

+kB ̄ 0 vz)ξ=0. (10.164)

On the vacuum side Eq.(10.162) gives

B ̄ 1 vr = |k|β

[

I|′m|Kˆ|′m|−Iˆ|′m|K′|m|

]

(10.165)

where omission of the argument means that the modified Bessel function is evaluated

atr=a.Substitution of Eq.(10.165) into Eq.(10.164) gives

β=

i

(m

a

+kB ̄ 0 z

)

|k|

[

I′|m|Kˆ|′m|−Iˆ|′m|K|′m|

]ξ (10.166)

so the complete vacuum field can now be expressed in terms ofξ.

3. Plasma side of plasma-vacuum interface: The plasma-side version of Eq.(10.163)
   gives
   B ̄ 1 pr−ikB ̄ 0 pzξ=0 (10.167)
   sinceB 0 pθvanishes inside the plasma. From Eq.(10.158) the perturbed radial mag-
   netic field on the plasma side of the interface is

B ̄ 1 pr=|k|αI|′m|. (10.168)

The plasma side version of Eq.(10.164) is thus

α=

ikB ̄ 0 pz

|k|I|′m|

ξ (10.169)

and so the plasma fields can now also be expressed in terms ofξ.

The stability condition, Eq.(10.156), can be written in terms ofαandβto obtain

β

(

im

a

+ikB ̄ 0 vz

)(

I|m|Kˆ|′m|−Iˆ|′m|K|m|

)

−

ξ

a

>αB ̄ 0 pzikI|m|. (10.170)

Substituting forαandβand re-arranging the order gives

|k|aB ̄^20 pz

[

I|m|

I|′m|

]

−

(m+kaB ̄ 0 vz)^2

|k|a

[

I|m|Kˆ|′m|−Iˆ|′m|K|m|

I|′m|Kˆ|′m|−Iˆ|′m|K′|m|

]

>1=⇒stable.(10.171)